Volume 114, Issue 5, 01 February 2001
 ARTICLES

 Theoretical Methods and Algorithms

Performance of CCSDT for first row diatomics: Dissociation energies and electron affinities
View Description Hide DescriptionCCSDT/augccpVXZ calculations were performed on the CN, and first row diatomic molecules. The inclusion of diffuse functions improves the dissociation energies of the anionic systems by 2.0–3.4 kcal/mol, which is relevant bearing in mind the goal of achieving chemical accuracy. The contribution of the diffuse functions in the case of neutral (0.6 kcal/mol) is by no means negligible in this context. A serious discrepancy between the theoretical prediction and the experimental values available for the dissociation energy of was found. Since the theoretical deficiences commonly ascribed to the CCSDT method (singlereference and spin contamination when using UHF zerothorder wave functions) cannot be invoked in this case, further experimental work is required to throw some light on the origin of such a discrepancy. The performance of CCSDT for adiabatic electron affinities is excellent in the case of and For the observed discrepancies can be explained in terms of the wellknown multiconfigurational nature of the ground state of the molecule.

Development of an efficient linear response approach to the Hilbert space multireference coupledcluster theory
View Description Hide DescriptionIn this paper, we use an analytic linear response to develop efficient expressions for calculating a firstorder energy response using the multireference Hilbert spacecoupledcluster (HSMRCC) theory. Equations for the firstorder response are derived and their diagrammatic evaluation is outlined. The Zvector formalism used in SRCC to eliminate the explicit presence of a cluster amplitude response in favor of a deexcitation operator is generalized to HSMRCC and applied here. We also discuss several aspects of the Zvector and outline different ways of introducing the technique and appropriateness of these in various circumstances. Efficient expressions for the energy response in terms of statedependent effective CC density matrices are presented. We also compare our approach with Szalay’s approach based on the generalized Hellmann–Feynmann theorem and discuss the advantages of our approach.

Electronimpact vibrational excitation of polyatomic gases: Exploratory calculations
View Description Hide DescriptionWe present model calculations for the inelastic cross sections of electron collisions with tetrahedral molecules Si, Ge) when only the molecular “breathing” mode, is being excited. The collision energy range is well above the excited thresholds and up to 12 eV, where the adiabatic approximation for the inelastic matrix is expected to hold. The results show the efficiency of the shape resonance in enhancing the inelastic process and the appearance, in the two heavier targets, of a further resonance in the inelastic channels of both molecules. The corresponding excitation rates are also computed together with estimates of the vibrational excitation functions.

Quantumclassical Liouville description of multidimensional nonadiabatic molecular dynamics
View Description Hide DescriptionThe quantumclassical Liouville formulation gives a quantummechanical densitymatrix description of the “quantum” particles of a problem (e.g., the electrons) and a classical phasespacedensity description of the “classical” particles (e.g., the nuclei). In order to employ this formulation to describe multidimensional nonadiabatic processes in complex molecular systems, this work is concerned with an efficient Monte Carlo implementation of the quantumclassical Liouville equation. Although an exact stochastic realization of this equation is in principle available, in practice one has to cope with two major complications: (i) The representation of nonlocal phasespace operators in terms of local classical trajectories and (ii) the convergence of the Monte Carlo sampling which is cumbersome due to complexvalued trajectories with rapidly oscillating phases. Several strategies to cope with these problems are discussed, including various approximations to determine the momentum shift associated with a nonadiabatictransition, the onthefly generation of new trajectories at curvecrossings, and the localization of trajectories after irreversible electronic transitions. Employing several multidimensional model systems describing ultrafast photoinduced electron transfer and internal conversion, detailed numerical studies are performed which are compared to exact quantum calculations as well as to the “fewestswitches” surfacehopping method. In all cases under consideration, the Liouville calculations are in good agreement with the quantum reference. In particular, the approach is shown to provide a correct quantumclassical description of the electronic coherence.

Improved trial wave functions in quantum Monte Carlo: Application to acetylene and its dissociation fragments
View Description Hide DescriptionRecent quantum Monte Carlo (QMC) studies of electronic structure have considered various trial function enhancements directed at improved fixednode energies. In this study we investigate complete active space selfconsistent field (CASSCF) trial functions in the diffusionMonte Carlo (DMC) method. We study longer CASSCF expansions than typically used in QMC studies and optimize correlation function parameters, basis function coefficients, and configuration state function mixing coefficients. To perform a stable, efficient wavefunction optimization, sample points are analytically obtained from an integrable probability density function or a Monte Carlo walk guided by a positive definite function. The approach is applied to acetylene and its dissociation fragments For these systems 70%–90% of the correlation energy is recovered with variational MC and 91%–98% with DMC.

New approach to the design of density functionals
View Description Hide DescriptionThe prevailing approach in densityfunctional theory makes use of universal functionals, valid for any number of electrons (N). In this article and following the work of Lieb [E. H. Lieb, Int. J. Quantum Chem. 24, 243 (1983)], we argue that the use of Ndependent functionals could be an equally valid approach. Size consistency(dissociation into proper fragments) puts restrictions on the form that Ndependent functionals may adopt. We propose a simple procedure for correcting the sizeconsistency problem of existing Ndependent functionals that also provides an original way of designing new ones.

Molecular dynamics analog of the reverse Monte Carlo method
View Description Hide DescriptionA new molecular dynamics algorithm is reported to evaluate data of diffraction experiments on liquids or amorphous materials. The atoms, in addition to the excluding shortrange forces, are moved in a fictitious potential field defined by the chisquare difference of the experimental structure factors. The results provided by the new reverse molecular dynamics algorithm are equivalent to those obtained using the reverse Monte Carlo method for the test model systems of the LennardJones liquid and water. We exploited the test calculations of the water system to discuss the separation capabilities of these modeling techniques. The results were encouraging, if the number of different (good quality) experimental functions as input information is larger or equivalent with the number of particle paircorrelation functions necessary to characterize the structure of the system. However, the results indicated dubious outcomes, if there were more partials estimated than different experimental data sets. This questions the conclusions of some previous studies.

Efficiently computing boundstate spectra: A hybrid approach of the multiconfiguration timedependent Hartree and filterdiagonalization methods
View Description Hide DescriptionWe present a new approach for determining boundstate spectra of molecules or clusters. In our approach a wave packet propagation is performed, which exploits the efficiency of the multiconfiguration timedependent Hartree scheme, to produce an autocorrelation function. From this, an accurate spectrum is extracted employing the filterdiagonalization procedure. The accuracy of this hybrid method is demonstrated by applying it to the spectrum of carbon dioxide. Compared with the filterdiagonalization scheme based on a numerically exact wave packet propagation and with a matrix diagonalization using the Lanczos algorithm, our approach turns out to be more efficient. The method can easily be generalized to the treatment of resonant states.

Irreducible Brillouin conditions and contracted Schrödinger equations for nelectron systems. I. The equations satisfied by the density cumulants
View Description Hide DescriptionTwo alternative conditions for the stationarity of the energy expectation value with respect to kparticle excitations are the kparticle Brillouin conditions and the kparticle contracted Schrödinger equations, These conditions express the kparticle density matrices in terms of density matrices of higher particle rank. The latter can be eliminated if one expresses the in terms of their cumulants but this is not sufficient to make the or separable (extensive), i.e., they are not expressible in terms of only connected diagrams. However, in a formulation based on the recently introduced general normal ordering with respect to arbitrary wave functions, the irreducible counterparts and of the and can be defined. They are easily evaluated explicitly in terms of the generalized Wick theorem for arbitrary wave functions, and they lead to equations for the direct construction of the cumulants which are additively separable quantities and which scale linearly with the system size. The or the are necessary conditions for γ and the to represent an exact nfermionic eigenstate of the given Hamiltonian. To specify the desired state, additional conditions must be satisfied as well, e.g., the partial trace relations which relate to γ and The particle number and the total spin must be specified and nrepresentability conditions enter implicitly. While the nondiagonal elements of γ and the are determined by the or the the additional conditions mainly serve to fix the diagonal elements. A hierarchy of kparticle approximations is defined. It is based on the fact that the expansion in terms of cumulants can be truncated at any particle rank, which would not be possible for the density matrices For closedshell states the oneparticle approximation agrees with Hartree–Fock.

Semiempirical models for image electrostatics. I. Bare external charge
View Description Hide DescriptionWe develop a simple semiempirical Pariser–Parr–Popletype model of a metal. It can be used for studies of adsorption and of the interactions between the metal surface and external charges, and is compatible with ab initio description of the molecules outside the metal. We investigate the dependence of the imagetype field on the few parameters that control the model behavior, and develop a simple direct minimization selfconsistent field technique that is useful for convergence in nearly degenerate cases.

The Rys quadrature revisited: A novel formulation for the efficient computation of electron repulsion integrals over Gaussian functions
View Description Hide DescriptionA novel formulation of the Rys quadrature algorithm for the calculation of the electron repulsion integrals over Gaussian basis functions is presented. The new algorithm is specifically designed for high contractions. As for the original Rys quadrature algorithm, the new algorithm is very efficient for high angular momentum functions. In addition it is also equally efficient for low angular momentum functions. The new algorithm takes unique advantage of (1) the numerical Rys quadrature methodology in (2) dealing with charge distributions a la McMurchie–Davidson and in (3) scaling integral blocks as a means of transferring angular momentuma la Gill–Head–Gordon–Pople. An analysis of the algorithm suggests very favorable floatingpoint operation counts.

A strategy for analysis of (molecular) equilibrium simulations: Configuration space density estimation, clustering, and visualization
View Description Hide DescriptionWe propose an approach for summarizing the output of long simulations of complex systems, affording a rapid overview and interpretation. First, multidimensional scaling techniques are used in conjunction with dimension reduction methods to obtain a lowdimensional representation of the configuration space explored by the system. A nonparametric estimate of the density of states in this subspace is then obtained using kernel methods. The free energysurface is calculated from that density, and the configurations produced in the simulation are then clustered according to the topography of that surface, such that all configurations belonging to one local free energy minimum form one class. This topographical cluster analysis is performed using basin spanning trees which we introduce as subgraphs of Delaunay triangulations. Free energysurfaces obtained in dimensions lower than four can be visualized directly using isocontours and surfaces. Basin spanning trees also afford a glimpse of higherdimensional topographies. The procedure is illustrated using molecular dynamics simulations on the reversible folding of peptide analoga. Finally, we emphasize the intimate relation of density estimation techniques to modern enhanced sampling algorithms.

Langevin stabilization of molecular dynamics
View Description Hide DescriptionIn this paper we show the possibility of using very mild stochastic damping to stabilize long time step integrators for Newtonian molecular dynamics. More specifically, stable and accurate integrations are obtained for damping coefficients that are only a few percent of the natural decay rate of processes of interest, such as the velocity autocorrelation function. Two new multiple time stepping integrators, Langevin Molly (LM) and Brünger–Brooks–Karplus–Molly (BBK–M), are introduced in this paper. Both use the mollified impulse method for the Newtonian term. LM uses a discretization of the Langevin equation that is exact for the constant force, and BBK–M uses the popular Brünger–Brooks–Karplus integrator (BBK). These integrators, along with an extrapolative method called LN, are evaluated across a wide range of damping coefficient values. When large damping coefficients are used, as one would for the implicit modeling of solvent molecules, the method LN is superior, with LM closely following. However, with mild damping of 0.2 ps^{−1}, LM produces the best results, allowing long time steps of 14 fs in simulations containing explicitly modeled flexible water. With BBK–M and the same damping coefficient, time steps of 12 fs are possible for the same system. Similar results are obtained for a solvated protein–DNA simulation of estrogen receptor ER with estrogen response element ERE. A parallel version of BBK–M runs nearly three times faster than the VerletI/rRESPA (reversible reference system propagator algorithm) when using the largest stable time step on each one, and it also parallelizes well. The computation of diffusion coefficients for flexible water and ER/ERE shows that when mild damping of up to 0.2 ps^{−1} is used the dynamics are not significantly distorted.

The Jacobi–Wilson method: A new approach to the description of polyatomic molecules
View Description Hide DescriptionWe present a new method adapted to the calculation of excited rovibrational states of semirigid molecules. It first relies on a description of the molecule in terms of polyspherical coordinates of Jacobi vectors, in order to obtain a compact expression for the kinetic energy operator This general description is then adapted to the molecule considered by defining curvilinearnormal modes from the corresponding zero order harmonic Hamiltonian the solutions of which are being used as the working basis set. The residual kinetic term is treated mainly analytically in this basis, and displays no radial contribution. Anharmonic coupling is handled by means of a pseudospectral scheme based on Gauss Hermite quadratures. This method is particularly adapted to direct iterative approaches which only require the action of on a vector, without the need of the associated matrix, thus allowing ultralarge bases to be considered. An application to the excited vibrational states of the HFCO molecule is presented. It is shown in this example that energy levels can be trivially assigned from the leading expansion coefficient of the associated eigenvector.

Ab initio adiabatic dynamics involving excited states combined with Wigner distribution approach to ultrafast spectroscopy illustrated on alkali halide clusters
View Description Hide DescriptionWe investigate the ultrafast multistate nuclear dynamics involving adiabatic electronic excited states of nonstoichiometric halide deficient clusters characterized by strong ionic bonding and one excess electron, which is localized either in the halide vacancy or on the alkali atom attached to the ionic subunit depending on the cluster size. For this purpose we developed an ab initio adiabatic nuclear dynamics approach in electronic excited and ground states “on the fly” at low computational demand by introducing the “frozen ionic bonds” approximation, which yields an accurate description of excited states considering the excitation of the one excess electron in the effective field of the other valence electrons involved in the ionic bonding. We combined this multistate dynamics approach with the Wigner–Moyal representation of the vibronic density matrix forming the ab initio Wigner distribution approach to adiabatic dynamics. This method allows the simulation of femtosecondNeExPopump–probe and NeExNepump–dump signals based on an analytic formulation which utilizes temperaturedependent groundstate initial conditions (Ne), an ensemble of trajectories carried out on the electronic excited state (Ex) for the investigation of the dynamics of the system, and either the cationic (Po) or the ground state (Ne) for the probing step. The choice of the systems has been made in order to determine the time scales of processes involving (i) metallic bond breaking such as during the dynamics in the first excited state of and (ii) fast geometric relaxation leaving the bonding frame intact as during the dynamics in the first excited state of The bondbreaking process via a conical intersection involving nonadiabatic dynamics will be presented in the accompanying paper [Hartmann et al., J. Chem. Phys. 114, 2123 (2001)]. The dynamics in the first excited state of from triangularto linearto triangular structure gives rise to fast geometric relaxation due to Na–Na bond breaking at the time scale of ∼90 fs but no signature of internal vibrational energy redistribution (IVR) is present in NeExNepump–dump signals since the broken metallic bond prevents the coupling between stretching and bending modes. Instead, anharmonicities of the bending periodic motion have been identified. In contrast, in the case of which is the smallest finite system for a surface Fcenter prototype of bulk color centers, after the geometric relaxation in the excited state of ∼100 fs leading to the deformed cuboidal type of structure without breaking of bonds, different types of IVR have been identified in NeExNe signals by tuning the dump laser: onemode selective energy leaving IVR, resonant, and restricted energy arriving IVR corresponding to the selection of different parts of the phase space. Dissipative IVR could not be identified in NeExNe signals of at low initial temperature on the time scale up to 2 ps in spite of 15 degrees of freedom. Due to similar structural and electronic properties such as F centers in bulk, these findings can serve as guidance for establishing the time scales for geometric relaxation and IVR in excited states of larger systems.

Ab initio nonadiabatic dynamics involving conical intersection combined with Wigner distribution approach to ultrafast spectroscopy illustrated on cluster
View Description Hide DescriptionWe present a theoretical study of a femtosecond photo isomerization process due to a nonadiabatic radiationless decay from the first excited state through a conical intersection occurring in one of the nonstoichiometric halidedeficient clusters with one excess electron This is an extension of the adiabatic dynamics study presented in the accompanying paper [J. Chem. Phys. 114, 2106 (2001)] for other members of the series characterized by a strong ionic bonding for which the “frozen ionic bonds” approximation has been justified, allowing consideration of the optical response of the single excess electron in the effective field of the other electrons. In this contribution we outline the extension of the ab initio Wignerdistribution approach to nonadiabatic molecular dynamics which combines the Wigner–Moyal representation of the vibronic density matrix with the ab initio multistate molecular dynamics in the ground and excited electronic states including the nonadiabatic coupling computed “on the fly” in connection with the fewestswitches hopping algorithm. This scheme allows accounting for temperaturedependent initial conditions, for the propagation in the excited state and in the ground state after the passage through the conical intersection, and for probing in the cationic ground state as well as for deriving analytic expressions for the pump–probe signals which utilize an ensemble of classical trajectories obtained at low computational demand. Our approach permits investigation of the photo isomerization through the conical intersection due to the long amplitude motion in the system in full complexity, taking into account all degrees of freedom. After breaking of one metallic and of one ionic bond the conical intersection occurs at the linear geometry and involves states of different symmetry which differ in the translocation of the one excess electron or positive charge localized at the Na atom from one end to the other of the system and separates two isomers with and structures. From the analysis of the nonadiabaticdynamics, the time scales for the metallic bond breaking of ∼90 fs and for the ionic bond breaking of ∼220 fs, for the passage through the conical intersection after ∼0.4 ps and for the internal vibrational energy redistribution (IVR) of more than 0.9 ps for the individual isomers, have been determined. The simulated fs pump–probe signals confirm the above results and provide the information about the experimental conditions such as laser frequencies and pulse duration under which bond breaking of different type as well as the population of each of the two isomers after the passage through the conical intersection can be identified. In this contribution we show that the mechanism of the photo isomerization at a conical intersection due to a long amplitude motion can occur in atomic clusters and is not necessarily limited to organic photochemistry.
 Gas Phase Dynamics and Structure: Spectroscopy, Molecular Interactions, Scattering, and Photochemistry

Observation of the FeNC molecule by laser fluorescence excitation spectroscopy
View Description Hide DescriptionA new molecular electronic band system has been observed in the near UV by laser fluorescence excitation in a freejet supersonic expansion of photolyzed diluted in seed gases containing acetonitrile. The three observed bands displayed the rotational structure of a transition and could be assigned as an excitedstate progression in a single vibrational mode. The rotational constant and its change upon isotopic substitution in acetonitrile was consistent with the assignment of the molecular carrier as the linear FeNC molecule. In analogy with the iron monohalides, which have electronic band systems in the same wavelength range, the transition can be described as a excitation. The derived structure is compared with that of other metalcontaining cyanide/isocyanide molecules.

Electronic structure of the and collision complexes: Experiment and theory
View Description Hide DescriptionThe electronic structure of the and collision pairs is probed by optical excitation. Based on polarization experiments, we measure the alignment tensor of the electronic transition dipole moments. We compare the experimental data with the results of a quantum chemical calculation. The different geometric properties of the electronic wave functions of the two collision pairs are directly visible in the experimental data.

The reaction from thermal to 25 eV
View Description Hide DescriptionThe endothermic reaction has been investigated both experimentally and by ab initio calculations. Integral cross sections are presented as a function of collision energy. For excited ions the reaction onset shifts towards lower energies, indicating that the internal energy of the reactant ions promotes the reaction. Analysis of the energy dependence of the cross section does not allow an unequivocal determination of the reaction endothermicity. However, calculations of the ground doublet surface indicate that the minimum energy path from reactants to products proceeds through strongly bent geometries, with an endothermicity equivalent to the thermodynamic value of about 5 eV. For linear geometries an additional barrier of about 1 eV is found in the product channel. The structure of in its first excited quartet state has been also calculated by ab initio methods.

The reactions and as a test of current direct dynamics computational methods to determine variational transitionstate rate constants. I.
View Description Hide DescriptionIn the present work, we have theoretically calculated the rate constants and their temperature dependence for the reactions and for the reaction of methane with OD, by means of variational transitionstate theory plus multidimensional tunneling corrections, at the MPSAC2//MP2/ccpVTZ/// and CCSD(T)//MP2/ccpVTZ/// electronic levels. Also, the newly developed singlepoint energy interpolation algorithm has been used at the CCSD(T)/augccpVTZ//MP2/ccpVTZ and CCSD(T)SAC//MP2/ccpVTZ levels. For reactions with 2 or 3, the competitive canonical unified statistical theory has been applied as they involve more than one nonequivalent reaction channel. Variational effects and tunneling have been found to be very important. The proton shift classical energy barrier turns out to be 5.83 and 4.97 kcal/mol at the CCSD(T)/augccpVTZ//MP2/ccpVTZ and CCSD(T)SAC//MP2/ccpVTZ levels, respectively. Even though we have used the highest ab initio electronic level reported up to now for dynamics calculations on these reactions, and although our results are quite good, we still do not match exactly the available experimental data. From our results it can be inferred that, probably, an adiabatic energy maximum between the CCSD(T)SAC//MP2/ccpVTZ and CCSD(T)/augccpVTZ//MP2/ccpVTZ values (5.6 and 6.2 kcal/mol, respectively, for the perprotio reaction) could be the most feasible, and that the description of the adiabatic profile fails especially in that region away from the transitionstate location but crucial for tunneling corrections.